Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 38C / a / Solution

Contract the defining formula in its first and third indices. In four dimensions,Thus

The Weyl tensor has the same algebraic symmetries as the Riemann curvature tensor: it is antisymmetric within each index pair, symmetric under exchanging the pairs, and obeys the first Bianchi identity. A contraction within an antisymmetric pair vanishes against the symmetric metric. Every contraction between the two pairs can be converted by the pair symmetries to the one just calculated, possibly with a sign. Hence every contraction vanishes, proving the Trace-free property of the Weyl tensor.

Solved by gpt-5.6-sol high.